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Incompleteness: The Proof and Paradox of Kurt Godel by Rebecca Goldstein

This is the story of Kurt Godel, considered to be the greatest logician since Aristotle and the second greatest mind of the 20th century after Einstein. The book includes discussion of the Vienna Circle, whose meetings Godel sometimes attended; Godel’s differences with Wittgenstein, whose views impressed the Vienna Circle but not Godel; and Godel’s friendship with Einstein, when both of them were at the Institute for Advanced Studies in Princeton.

Godel is most famous for proving that that any formal system that is rich enough to contain arithmetic or number theory must contain a true statement that can neither be proven nor disproven, in other words, that such a system is necessarily incomplete. A corollary of the incompleteness theorem is that any such system cannot be proven to be consistent within the system itself. Goldstein suggests that the incompleteness theorem demonstrates that there is a mathematical reality beyond the reach of any human-made formal system.

Goldstein steps through the proof of incompleteness, but I didn’t or couldn’t follow the whole proof. The proof seems to rely on the strange consequences that result from sentences that refer to themselves, such as “This sentence is false”. Russell ruled out such sentences in his Theory of Types, and that seems like a good idea to me. It may be arbitrary, but it seems right to say that sentences that refer to themselves and sets that contain themselves should not be allowed in any formal system, given the paradoxical results that follow. (5/11/10)